I am back with another great exercise, however I am again a little bit stuck on proving it further, it goes as following:
Suppose $A$ and $B$ are both non empty, disjoint Sets, with $|A|=2n-1$ and $|B|=2n-1$. The Powersets of $A$ and $B$ are $P(A)$ and $P(B)$. Show that: $$|P(A) \cup P(B)| = {2n \choose n}$$
And as for a hint it was noted that „It might be helpful to find a bijection between the union of both Powersets and a given set with the cardinality of 2n choose n”
As for my solution I think I have found a general approach on how to solve this equality; yet I am not sure on how to proceed further to show that it does apply for the cardinality of both sets in specific and if there might be a better solution on how to do it. Regardless, here is my approach. Generally I know that the cardinality of an non empty Powerset for example the powerset of $X$ would be: $$|P(X)| = 2^{|X|}$$ and if $X$ would have $n$ elements it would be $2^{n}$. Knowing this I could rewrite the equality with help of the binomial theorem which would give for $$2^{n} = (1+1)^{n} = \sum_{k=0}^n {n \choose k} * 1^{n-k}1^{k} = \sum_{k=0}^n {n \choose k}$$
Further now for the Powersets of $A$ and $B$ I know that: $$|P(A) \cup P(B)| = 2^{|A|+|B|} = 2^{|A|}2^{|B|} \le |P(A\cup B)| = 2^{|A|} + 2^{|B|} $$ which I used in order to show with the binomial theorem, if I would suppose that both Powersets have only n elements: $$\begin{align}|P(A) \cup P(B)| =& 2^{|A|+|B|} = 2^{|A|}2^{|B|} = |P(A)|\cdot|P(B)|\\=& \sum_{k=0}^n {n \choose k} \cdot\sum_{i=0}^n {n \choose i} =\sum_{k=0}^n {n \choose k} \cdot {n \choose n-k} = \sum_{k=0}^n {n \choose k}^{2}\\=&(1+1)^{n}(1+1)^{n} = (1+1)^{2n} = 2^{2n}\end{align}$$ hence $${2n \choose n}$$
However I as I am rereading my answer, I have some doubts on my reasoning and on it’s correctness and further am trying to find a better way which will also include the actual cardinalities for both Powersets (and how could I establish a connection with a bijection according to the hint ? I know that there is a certain proof on multi-sets but how can I relate it to this exercise?). I would once again be very very grateful for all help, suggestions and advice and any sort of input; I still have a lot to learn, thank you for having me :)